Editing Orthogonal matrix (section)

==Elementary constructions==

===Lower dimensions===
The simplest orthogonal matrices are the {{nowrap|1 × 1}} matrices [1] and [−1], which we can interpret as the identity and a reflection of the real line across the origin.

The {{nowrap|2 × 2}} matrices have the form
<math display="block">\begin{bmatrix}
p & t\\
q & u
\end{bmatrix},</math>
which orthogonality demands satisfy the three equations
<math display="block">\begin{align}
1 & = p^2+t^2, \\
1 & = q^2+u^2, \\
0 & = pq+tu.
\end{align}</math>

In consideration of the first equation, without loss of generality let {{math|1=''p'' = cos ''θ''}}, {{math|1=''q'' = sin ''θ''}}; then either {{math|1=''t'' = −''q''}}, {{math|1=''u'' = ''p''}} or {{math|1=''t'' = ''q''}}, {{math|1=''u'' = −''p''}}. We can interpret the first case as a rotation by {{mvar|θ}} (where {{math|1=''θ'' = 0}} is the identity), and the second as a reflection across a line at an angle of {{math|{{sfrac|''θ''|2}}}}.

<math display="block">
\begin{bmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta \\
\end{bmatrix}\text{ (rotation), }\qquad
\begin{bmatrix}
\cos \theta & \sin \theta \\
\sin \theta & -\cos \theta \\
\end{bmatrix}\text{ (reflection)}
</math>

The special case of the reflection matrix with {{math|1=''θ'' = 90°}} generates a reflection about the line at 45° given by {{math|1=''y'' = ''x''}} and therefore exchanges {{mvar|x}} and {{mvar|y}}; it is a [[permutation matrix]], with a single 1 in each column and row (and otherwise 0):
<math display="block">\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}.</math>

The identity is also a permutation matrix.

A reflection is [[Involutory matrix|its own inverse]], which implies that a reflection matrix is [[symmetric matrix|symmetric]] (equal to its transpose) as well as orthogonal. The product of two rotation matrices is a [[rotation matrix]], and the product of two reflection matrices is also a rotation matrix.

===Higher dimensions===
Regardless of the dimension, it is always possible to classify orthogonal matrices as purely rotational or not, but for {{nowrap|3 × 3}} matrices and larger the non-rotational matrices can be more complicated than reflections. For example,
<math display="block">
\begin{bmatrix}
-1 & 0 & 0\\
0 & -1 & 0\\
0 & 0 & -1
\end{bmatrix}\text{ and }
\begin{bmatrix}
0 & -1 & 0\\
1 & 0 & 0\\
0 & 0 & -1
\end{bmatrix}</math>

represent an ''[[Inversion in a point|inversion]]'' through the origin and a ''[[improper rotation|rotoinversion]]'', respectively, about the {{math|z}}-axis.

Rotations become more complicated in higher dimensions; they can no longer be completely characterized by one angle, and may affect more than one planar subspace. It is common to describe a {{nowrap|3 × 3}} rotation matrix in terms of an [[axis and angle]], but this only works in three dimensions. Above three dimensions two or more angles are needed, each associated with a [[plane of rotation]].

However, we have elementary building blocks for permutations, reflections, and rotations that apply in general.

===Primitives===
The most elementary permutation is a transposition, obtained from the identity matrix by exchanging two rows. Any {{math|''n'' × ''n''}} permutation matrix can be constructed as a product of no more than {{math|''n'' − 1}} transpositions.

A [[Householder reflection]] is constructed from a non-null vector {{math|'''v'''}} as
<math display="block">Q = I - 2 \frac{{\mathbf v}{\mathbf v}^\mathrm{T}}{{\mathbf v}^\mathrm{T}{\mathbf v}} .</math>

Here the numerator is a symmetric matrix while the denominator is a number, the squared magnitude of {{math|'''v'''}}. This is a reflection in the hyperplane perpendicular to {{math|'''v'''}} (negating any vector component parallel to {{math|'''v'''}}). If {{math|'''v'''}} is a unit vector, then {{math|1=''Q'' = ''I'' − 2'''vv'''<sup>T</sup>}} suffices. A Householder reflection is typically used to simultaneously zero the lower part of a column. Any orthogonal matrix of size {{nowrap|''n'' × ''n''}} can be constructed as a product of at most {{mvar|n}} such reflections.

A [[Givens rotation]] acts on a two-dimensional (planar) subspace spanned by two coordinate axes, rotating by a chosen angle. It is typically used to zero a single subdiagonal entry. Any rotation matrix of size {{math|''n'' × ''n''}} can be constructed as a product of at most {{math|{{sfrac|''n''(''n'' − 1)|2}}}} such rotations. In the case of {{nowrap|3 × 3}} matrices, three such rotations suffice; and by fixing the sequence we can thus describe all {{nowrap|3 × 3}} rotation matrices (though not uniquely) in terms of the three angles used, often called [[Euler angles]].

A [[Jacobi rotation]] has the same form as a Givens rotation, but is used to zero both off-diagonal entries of a {{nowrap|2 × 2}} symmetric submatrix.